To Shlomo Vinner, with many thanks for bringing the question about “the meaning of it all” into the field of mathematics education.
AFFECTIVE TRANSGRESSION AS THE CORE OBJECTIVE OF MATHEMATICS EDUCATION
Barbara Pieronkiewicz
Pedagogical University of Cracow,
Basiapieronkiewicz @ o2.pl
Introduction
Some people, not necessarily students, consider mathematics as strongly related to many other fields of knowledge or even fields of human thought and action. Mathematics has its unquestionable roots in philosophy, and for many centuries they both have occupied the same ground. Similarly, one cannot deny its relation to physics, economics, and other branches of science where either computation or logical reasoning play important role. Somewhat surprisingly, in light of Koestler’s (1964) theory of bisociation, mathematics has also a natural connection with Humour and Art. Despite the fact that some mathematicians have tried to show “the human face of mathematics” (White, 1993) and wide spectrum of literary, pedagogical, psychological, and sociological aspects of mathematics, such claims remain definitely in a minority position.
In this paper, mathematics education, which is by definition an interdisciplinary field of knowledge, is considered as marriage of mathematics to the meaning of life (i.e. Vinner, 2013) and the essence of the human being as such. Moreover, within this particular domain, linking two cultures: the sciences and the humanities, distinguished significantly by C.P.Snow (1959) is not only possible, but simply necessary. For this reason, mathematics educators pose questions concerning not only mathematics, but also cognitive and affective basis of human’s thinking, feeling and behaviour.
It is commonly known that mathematics as a school subject often raises a lot of negative emotions within students. It is seen as a difficult, detached from reality, full of useless in everyday life definitions and theorems field of knowledge. Not surprisingly then, it causes a lot of anxiety, emotional tension and internal discord. Nowadays many students declare their humanities preferences in purpose to justify the lack of involvement in learning mathematics. That is why a considerable part of teachers’ efforts are focused on how to bring about desired change in students’ dealings with mathematics at school. It seems, however, that the process of change, understood as an alteration of cognitive and affective structures, has not gained sufficient researchers’ attention so far. This paper is a modest attempt to explore this phenomenon occurring within the affect domain, in relation to the underachievement syndrome and a general problem of how people change.
Keywords: transgressive concept of a man, affective transgression in learning mathematics, underachievement syndrome
Underachievers: the Ugly Ducklings of mathematics education
A lot of attention has been paid to students’ achievement in mathematics so far. Although researchers and teachers’ efforts seek to improve the quality and efficiency of mathematics education, there are still many students who – for several reasons – achieve low scores in this school subject. What is more unsettling, many students declare humanities preferences in order to explain and legitimize their lack of engagement for learning mathematics and even among those who reveal their potential, there might be a significant number of underachievers (Rimm, 2008). According to some dictionary definitions, the underachiever is:
a student who performs less well in school than would be expected on the basis of abilities indicated by intelligence and aptitude tests, etc.
or
a person that performs below expectations (Dictionary.com Unabridged)
but also
someone (such as a student or athlete) who does not perform as well or work as hard as he or she can (Merriam-Webster.com)
According to Rimm’s definition:
Underachievement is a discrepancy between a child's school performance and some index of the child's ability. If children are not working to their ability in school, they are underachieving. (Rimm, 1997, p. 18)
Understated student achievement may be temporary or chronic. In most cases, circumstantial decrease in performance transforms over time into a chronic one (Dyrda, 2007). If we take into account the range of Underachievement Syndrome, we can distinguish two types of the syndrome: local - related to one or two school subjects (subject specific underachievement) and global - referred to all school subjects (general underachievement).
In related literature, causes of the syndrome are usually categorized into three groups (i.e. Dyrda, 2000, 2007; Rimm, 1994, 1995):
1. Student dependent - reasons inherent in the learner; mental and somatic characteristics of the student: level of intellectual and emotional dysfunctions, physical defects, illnesses, but also lack of internal motivation, fear of failure and low self-esteem, which are considered to be the most decisive factors.
2. Environmental – social, economic and cultural conditions of the family, emotional environment within the family, parenting style, impact of group of peers;
3. School dependent - including poor organization of lessons, the lack of individualization of education, lack of additional classes, incompetent compensatory. Another important problem here is classifying students with the underdog stigma, which acts as a self-fulfilling prophecy (Golem effect). Stigmatized in such a way students easily fall into apathy and loose motivation. Designated by the authority of the teacher to the group of underdogs, also write themselves off.
Many research reports allow us to outline some attributes and behaviours typical to underachievers (Dyrda 2000; Rimm, 1994, 1995). Despite of low-level test outcomes and difficulties in completing tasks, these students formulate non-trivial questions and hypotheses. Although they put minimal effort to work in the classroom, they often have broad extracurricular interests. Behaviourally, they are described in terms of selective focusing attention, tendency to withdraw oneself (or to the contrary: offensive domination), hyperactivity (in extreme cases: including ADHD symptoms) and negative attitude to school duties. Most underachievers lack motivation for learning and they are bored with school. They set themselves too low or too ambitious goals.
Rimm (1997) adds to this portrait a following description:
Underachievers don't have internal locus of control, nor do they function well in competition. The lack of internal locus of control translates to a missed connection between effort and outcome; underachievers haven't learned about hard work. They know they're smart because they've been told that by almost everyone. (…) They just don't know how to be productively smart. If they put forth effort, they no longer have an excuse to protect their fragile self-concepts. They've defined smart as "easy," and anything that is difficult threatens their sense of being smart. (…) Underachievers habitually back away from losses and, therefore, don't build the resilience to cope with losing experiences, to see them as temporary. They feel permanently labelled "loser" if they don't do as well as others. (Rimm, 1997, p. 19 – 20)
In light of the above description, it seems reasonable to distinguish between two different strains of underachievement in learning mathematics. The first one would be related to recognized as highly gifted students who are able to gain excellence in this subject, yet they don’t achieve their full potential, whereas the second variant of underachievement should pertain to so called “humanists” who refuse to get involved in doing mathematics. In case of the latter, it is worth to note that many students hold the belief that to be good at mathematics, one needs to have some innate predispositions to grasp it, and that ordinary students cannot be expected to understand this school subject (Schoenfeld, 1992). In contrast, some psychologists or educators try to convince the broad audience to the egalitarian nature of mathematics. Already in 1973, Piaget mentioned that fact:
Any normal student is capable of correct mathematical reasoning, if attention is directed to activities of his interest, and if by this method the emotional inhibitions, that too often give him a feeling of inferiority in lessons in this area are removed. (...) There is no field where the "full development of the human personality" and the mastery of the tools of logic and reason which insure full intellectual independence are more capable of realization. (Piaget, 1973, pp. 98-99, 105)
Aligned with those thoughts, also Krygowska referred to the nature of mathematics:
There are different levels of mathematical activity and with the exception of extreme cases, we can find an appropriate level of activity to any normal student. (...) When it comes to the development of mathematical thinking, we must not write any student off. (...) The student has to take a fancy to mathematics, find pleasure in solving mathematical tasks, even though it requires effort and difficult concentration. (Krygowska, 1975, p243)
Although these statements were given in purpose to emphasize that mathematical skills can be developed by any student, they also lead us to affect and explicitly ascertain that it impinges on learning mathematics. It is reasonable to expect that students who are afraid to expose their emotional or intellectual shortcomings will choose to avoid doing mathematics (holding the belief that they are not able to learn it). Many students tend to have a distracting behaviour in classrooms in order to deviate their colleagues and teacher attention from their own mathematics difficulties. As part of my research, I have interviewed a high school student having low scores in mathematics. According to the information given by his teacher, the boy usually had side conversations, made sniping remarks and was playing around on every lesson. In an individual talk we had, this student confessed that he was showing off to get the rest of the class laughing. The real reason why he was doing that, was that he wanted to divert his classmates’ attention from his mathematical misunderstanding. He said that it seemed easier to undergo an imparted (thus somehow controlled) laughter on the jokes he made, than to hear his colleagues laughing on his failure. There is a general agreement amongst mathematics educators that, regarding students’ achievement, neither affect nor cognition should be underestimated (e.g. McLeod, 1992; Vinner, 2013). In that sense, many researchers have been examining how affect influences the field of mathematics education (e.g. DeBellis & Goldin, 1997, 2006; Schlöglmann, 2005). Fortus (2014) goes even further, stating that all science educators ought to pay more attention to affect. He claims there is a great urgency to lift students from boredom and indifference, because “without engagement, learning is partial, at best”. The example given above, shows clearly that if we want to convince young people, especially reluctant and disobedient low-achievers, that they may become ‘good at math’, we need to take into account their vulnerability to affective stimulation coming from mathematics. Among many features this discipline has, one of the most remarkable is that mathematics, like no other field of knowledge, evokes human’s affective responsiveness. For example, in the scope of the theoretical review one can find many references to math anxiety, and only very few references to anxiety outside mathematics (e.g. “chemistry anxiety” or “biology anxiety”). Nobody likes being frustrated, feeling fear or helplessness. Paradoxically, all of these negative experiences related to mathematics, could be seen as a positive phenomenon. As emphasized by DeBellis and Goldin (2006), teaching mathematics does not have to be necessarily focused on how to eliminate frustration and negative emotions. By developing meta-affective competences, both teacher and student may become able to experience negative emotions as something positive. What is even more important, all students in the classroom are exposed to similar trigger events, and the difference between low-achievers and successful learners is embedded in the way they cope with what they experience. In this context, it is the teachers’ responsibility to develop students’ meta-affective competences helping them to see their experiences through very different lenses. It is no surprise that if meta-affect is not encompassed by the established classroom environment, some students prefer to avoid mathematics, rather than take the challenge and confront constraints naturally submerged in every learning process.
In recent years, educators and teachers have put emphasis on students with special educational needs. According to Warnock Report (1978), in which the problem has been raised for the first time, there are many children who need professional support to meet the requirements of core curriculum. As a UNESCO member, Poland adapts its educational low to the recommendations of Salamanca Conference (1994). Implementation of the principle of inclusion is much concerned on students detected for some developmental deficits, mentally or physically handicapped and also the gifted ones. Those who perform below expectations and work less hard than they can, however, still remain in the shadow of public attention. Underachievement in mathematics, especially in case of secondary – school students inevitably has a ripple effect on academic learning. What is more, it prevents learners from applying for admission to sought-after fields of study and facilitating better job opportunities. Maree (2010) and Bentley (1998) pay attention to the harmful social, economical consequences of that issue: unemployment, social exclusion, unsuccessful relationships and marriages. The number of people who are unhappy and unsuccessful in their careers is probably proportional to the number of inadequate school achievers. Chronic school failure leads straight to underachievement in life. Thus, again, the problem faced by mathematics educators is more than just about students’ outcomes related to mathematics. It is about helping students to develop their potential and lead fulfilled lives. In order to ensure conditions for growth – promoting environment in the classroom, both researchers and teachers are called to elaborate on effective tools helping inclusively all students to overcome those affective and cognitive obstacles they encounter on the way towards widely understood success.
It is thus, of fundamental importance to focus research on understanding which agents play a key role in triggering underachievement onset, what we can do to both prevent and reverse underachievement syndrome in learning mathematics and how we can break the vicious circle of disaffection, underachievement and disengagement. Considering the transgressive concept of a man within mathematics education, brings ideas that may shed some new light on these issues.
Transgressive concept of a man
The term transgression is defined in different contexts (e.g. geology and genetics). In geology transgression is the spreading of the sea over land as evidenced by the deposition of marine strata over terrestrial strata; in genetics it means a peculiar case of heterosis - the increase in growth, size, fecundity, function or other characters in hybrids over those of the parents. In its transposition into psychological ground, Kozielecki (1987) uses terms of an intentional and deliberate overcoming of physical, social or symbolic boundaries. The concept of psychological transgression is devoted to the importance of the role that crossing over personal boundaries and subverting limitations play in everyone’s life. From this standpoint, a man is a self-directed, expansive creature who intentionally crosses the boundaries understood as demarcation lines separating what he is and what he owns, from what he may become.